I got a PM asking me which suit was best on PokerStars, after having done my other work analyzing monotonic-suit and paired-tripled flops in this thread.

I tested 215,108 hands, for which I saw 118,422 flops. In this sample, I saw:

88,485 Spades: 24.906%
88,662 Clubs: 24.957%
88,724 Diamonds: 24.974%
89,395 Hearts: 25.163%
---------------
355,266 total

As you can clearly see, hearts are clearly favored over spades. In fact, they are 0.26% MORE likely to appear on the flop (I have not yet conducted a turn/river study), and therefore a favorite.

Translated to $5/10 NL, this converts directly to $0.026 cents per hand. Over a sample of 215,108 hands (my sample above), this is a net earnings of $5,510.

Conclusion: Young players take note. Operations clearly indicate you should play hearts over spades. Understanding the statistics is important. Real data is more important than statistical models. Even with perfect understanding of statistics, things do not change. Always use caution. Learn to accept reality. Live with the reality you've been given. Bartering with sites does not change statistics. Entropic shufflers are possibly flawed. I would be careful reading this. Never take a conclusion for face value. Governing bodies such as the KGC may not be aware of these skews. Logging in and out will not change the empirical data. Energetic emails will go unheeded. Vary your play but stick to hearts over spades. Earnings will go up. Life tilt will go down. Ending this conclusion, it is important to understand one thing. Don't believe what you read.

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[1] Source code and intermediate files (my "flops" file which contains all flops) are available upon request. I'll even post in this thread.

[ ] sample size

100k hands for 4 possibilities is a sufficient sample size. The equivalent for 2 possibilities (a coinflip) would be 50,000 trials.

Would 50,000 trials be an insufficient sample size for flipping a coin?

this is some great info. ive always favored spades, especially the ace of spades b/c of motorhead. surely this data will help my winrate go up though

ive always wondered about things like this but i have a feeling it will prob be brushed off as zomg rigged.

YOU HAS DOCUMENTS???????/

So, let me get this straight, you are assuming based on those numbers that the true population proportions of those 4 suits are the 4 numbers you found? By that same logic if I flip a coin 50k times and get 24750 heads, does that mean the new probability to get heads is <50%? You're taking findings for fact where you should be just trying to show that P(getting any one suit) /= .25.

Hmm, a more useful number would be the statistical significance of such findings. Is this really that uncommon, I'm not sure. I doubt it.

The poker sites have done extensive research that indicated that fish prefer hearts, LDO.

Your results are biased.

There's not as many spades on the flop because players are much more likely to play spades than hearts, so a lot of the spades are already in players' hands.

The ace of spades is the prettiest card in the deck some say.

I'm not going to be so silly as to post 10 hands and say something as to the shuffle. I do have these hands, however, if you want them (and have a way to get them - the file is 2 megs, but contains hand history numbers). Here's a quick snippet if you want to check the hands (by downloading them yourself from Stars):

etc.

Actually, my results are unbiased - all I've done is run statistical analysis on a large sample of hands. You say it yourself, players are much more likely to play spades than hearts, so your odds of hitting a flush while holding hearts are better than while holding spades (slightly so, but the effect is there).

A++ post, especially now that I fully understand the conclusion.

Given my plug into calculator math, I find these results to be within the realm of possibility. A p-hat of .2516 with p knot = .25 and n = 355k isn't that significant. The resulting p-value is probably > 5%. And that one strays the most.

Statistics isn't really my area of expertise, so at the risk of sounding stupid I won't make any outlandish claims, but this seems perfectly normal. Take it to the math/prob forum, they can probably convince you.

You're almost right but you're doing it backwards. Our null hypothesis (h-naught) is that pokerstars deals an even distribution of hands. Therefore, our assumptive hypothesis (h-sub-a) is that pokerstars DOES deal more hearts than spades. Plugging this into JMP (my statistical analysis program) yields a p-value of 0.04982 - very very close to 0.05, typically defined as statistically significant. That is to say, the assumptive hypothesis is true, given our data, over 95% of the time (a statistically meaningful number for our values of n).

If you want, send me a PM and I can walk you through my steps (or I'd be happy to do them in this thread if you don't mind boring people with minutia of statistical analysis).

For what it's worth, I make no claims as to what this data MEANS or WHY it exists the way it does, I simply present it for your analysis.

My own personal thoughts are what 1p0kerboy said above - that more people are biased towards spades (which indicates the low number of spades being played) and biased AGAINST hearts (perhaps they are subconsciously feminine?).

What? No. I use the word knot to indicate a subscript 0 in the case that our null hypothesis is that p = .25 for hearts. It doesn't really matter how hearts compare to spades in this case. It is how hearts (actual) compares to hearts (predicted).

Now, that's almost the same p-value I got (.049995823). And I don't see where you get this idea that .05 is considered statistically significant; that depends entirely on the case. You would have more leniency in some cases and less in others. For example, you tighten your ME on like dimensions of a screw for a space shuttle, because your room for making a mistake is much narrower due to cost, time spent etc.

That p-value doesn't prove anything to me, I would need another sample first before I conclude anything. It'd be insane to generalize an entire distribution to one case that's borderline significant.

The best word to accurately describe this is inconclusive.

Also, when you make a claim like this, these types of statistical tests should be in your OP. Your argument is paper thin when you just throw out counts and then try to generalize to a whole distribution. That's just begging to be labeled as a ZOMG RIGGED! thread.

I understand what you're saying, but in general a p of <0.05 is considered "OK". You're absolutely right, for applications with .. let's call it "more negative utility" out of false positives and false negatives (alpha and beta), there are problems with choosing a p value as lax as 0.05.

For sitting around the forums and posting theories as to the occurrence of one suit over another in a game played by humans (where human tendencies and emotions DO affect the game), I suspect a p-value of 0.05 is more than accurate. Besides, we're not trying to sit around and convince ourselves that hearts are somehow better than spades/clubs/diamonds - what we're doing is analyzing a large sample and trying to see what ACTUALLY HAPPENS.

The fact of the matter is that people play less hearts than spades/clubs/diamonds, therefore hearts are the best suited hand to hold preflop.

I have just run the test on WCG|Rider's hand histories, and the same conclusion can be drawn:

Spades: 141303: 24.94%
Clubs: 141488: 24.98%
Diamonds: 141752: 25.03%
Hearts: 141818: 25.04%

This is over 188,787 flops, over 300k hands. Again, the bias is small BUT IS PRESENT AND IN EXACTLY THE SAME ORDER as in my sample. If I added these together, the bias would be even more pronounced.

Fact: Hearts are more common on the flop (over 500k hands!) than Diamonds, which are more common than Clubs, which are more common than spades.

red cards flop more than black cards?? racist ban imo

This really doesn't help your case. Comparing two suits to each other is a comparison of no utility, which I explained before but whatever, they are both statistics as opposed to parameters, and are thus unpredictable.

EDIT: Using that data, our p-value jumps all the way up to near .5. This only shows my point that using one sample to generalize about a distribution is a poor idea.

This is not true, and also there are problems with adding two samples like that to make one sample. If the first sample was bad, that doesn't make sense to add them, unless you are trying to make manipulate the proportions. Plus they are no longer random at that point.

All you are doing by adding them is trying to artificially increase n for the first sample.

CONCLUSION

Poker players are macho. Playing hearts is ghey. Playing spades is manly.

great, now do it over 10M hands.

this

As much as threads like this tilt me, it's good this is here to stop that. Well said.

I mean he says

and then

I mean he is making a huge claim, and then just throwing hands at us. The burden of proving this is on the OP.